Problems In Birational Geometry And Dynamical Systems

One of the core goal of algebraic and arithmetic dynamics is the explicit birational geometrical classification of algebraic varieties over an algebraically closed field(in arbitrary characteristic)and a number field under the viewpoint of dynamical systems[3,10,21,37,56,96,121,126].The main results of the thesis are the classification problem of projective varieties with automorphism groups of positive entropy,the derived length of automorphism groups with zero entropy acting on projective varieties and the study of the Kawaguchi-Silverman conjecture[62].T.-C.Dinh and N.Sibony[33]first proved that any abelian subgroup G of the automorphism group with positive entropy of a compact Kahler manifold M is free abelian of rank dr(G)?dim M-1.More generally,D.-Q.Zhang[122]and F.Hu[50]proved a theorem of Tits[113]type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic.In particular,any abelian subgroup G of the automorphism group with positive entropy of a projective variety is free abelian of rank ?dim X-1.In the last years,the maximal dynamical rank case r=dim X-1 has been intensively studied by D.-Q.Zhang in his series papers[122,124,125].We aim to characterize the pair(X,G)such that rank G=dim X-2.T.-C.Dinh,K.Oguiso and D.-Q.Zhang[31]established the derived length of zero entropy groups acting on a compact Kahler manifold of dimension n is at most n-1.We extends this result to projective varieties over an algebraically closed field of arbitrary characteristic.Meanwhile,we give a Fujiki-Lieberman type theorem in arbitrary characteristic.The Kawaguchi-Silverman conjecture(KSC for short)asserts that for a dominant self-map f:X?X of a projective variety X over (?) the arithmetic degree ?f(x)of any point x with Zariski dense f-orbit is equal to the first dynamical degree d1(f)of f.We assume that f is a morphism,and then reduce KSC to the following three cases:weak Calabi-Yau varieties,rationally connected varieties and varieties admitting a non-trivial special MRC fibration.In particular,when f is an automorphism and dim X=3,we have a more explicit result.Finally,we prove KSC holds for any endomorphism of any projective bundle over a smooth Fano variety with Picard number one.